On isometry and equivalence of skew constacyclic codes

In this paper we generalize the notion of n-isometry and n-equivalence relation introduced by Chen et al. in [13], [12] to classify constacyclic codes of length n over a finite field $F_{{}_q}$, where $q=p^r$ is a prime power, to the case of skew constacyclic codes without derivation. We call these relations respectively $(n,\sigma )$-equivalence and $(n,\sigma )$-isometric relation, where n is the length of the code and σ is an automorphism of the finite field. We compute the number of $(n,\sigma )$-equivalence and $(n,\sigma )$-isometric classes, and we give conditions on λ and μ for which $(\sigma ,\lambda )$-constacyclic codes and $(\sigma ,\mu )$-constacyclic codes are equivalent. Under some conditions on n and q we prove that skew constacyclic codes are equivalent to cyclic codes by using properties of our equivalence relation introduced. We also prove that when q is even and σ is the Frobenius automorphism, skew constacyclic codes of length n are equivalent to cyclic codes when $\gcd (n,r)=1$. Finally we give some examples as applications of the theory developed here.